Borgefors, G., Ragnemalm, I., di Baja, G.S.: The Euclidean distance transform: finding the local maxima and reconstructing the shape. In: Proc 7th Scand. Conf.
Filtering Non-Significant Quench Points Using Collision Impact in Grassfire Propagation Dakai Jin1(), Cheng Chen1, and Punam K. Saha1,2 1
Department of Electrical and Computer Engineering, University of Iowa, Iowa City, USA {dakai-jin,heng-chen,punam-saha}@uiowa.edu 2 Department of Radiology, University of Iowa, Iowa City, USA
Abstract. The skeleton of an object is defined as the set of quench points formed during Blum’s grassfire transformation. Due to high sensitivity of quench points with small changes in the object boundary and the membership function (for fuzzy objects), often, a large number of redundant quench points is formed. Many of these quench points are caused by peripheral protrusions and dents and do not associate themselves with core shape features of the object. Here, we present a significance measure of quench points using the collision impact of fire-fronts and explore its role in filtering noisy quench points. The performance of the method is examined on three-dimensional shapes at different levels of noise and fuzziness, and compared with previous methods. The results have demonstrated that collision impact together with appropriate filtering kernels eliminate most of the noisy quench voxels while preserving those associated with core shape features of the object.
1
Introduction
Skeletonization provides a compact yet effective representation of an object while preserving its important topological and geometrical features; see [1,2] for through surveys. Most of the popular skeletonization algorithms [2,3] are based on simulation of Blum’s grassfire propagation [4], where quench points are formed when two or more fire fronts collide and the skeleton is constructed from the set of these quench points. A well-known challenge with skeletonization is that small protrusions and dents on an object boundary create noisy quench points leading to noisy skeletal branches. This challenge is further intensified for fuzzy objects, because local maxima as well as ridges on the membership function create additional noisy quench points. Thus, the skeleton formed by the initial set of quench points consists of a large amount of redundant structures most of which carry little information related to core shape features of the object. Therefore, it is imperative to filter and remove less significant or noisy quench points to produce meaningful skeletons. This paper presents a new filtering algorithm to remove noisy quench points using the collision impact of Blum’s grassfire-fronts. Quench points have been defined and popularly used in skeletonization in the form of centers of maximal balls (CMB) [5]. CMB can be effectively identified in digital objects as the singularity points [5-7] in the distance transform (DT) map [8,9]. Arcel© Springer International Publishing Switzerland 2015 V. Murino and E. Puppo (Eds.): ICIAP 2015, Part I, LNCS 9279, pp. 432–443, 2015. DOI: 10.1007/978-3-319-23231-7_39
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li and Sanniti di Baja [5] introduced a criterion to detect the centers of maximal balls (CMBs) from a 3 3 neighborhood in integer-weighted distance transform, and Borgefors [7] extended it to 5 5 5 neighborhood. Saha and Wehrli [10] generalized the CMB for fuzzy objects, which was further studied by Svensson [11] where the fuzzy distance transform (FDT) [12] is used instead of DT to locate CMBs. Although, a few works [13-16] have been reported in literature to detect noisy or less significant quench points, a comprehensive theoretical formulation for characterization of significance of quench points is yet to emerge. Saha et al. [13] characterized surface- and curve-like shape points and recommended different support kernels to distinguish between noisy and significant quench points. Borgefors and Nyström [14] proposed a CMB reduction algorithm, where a CMB is marked as redundant if the maximal ball centered at it is covered by the union of some other maximal balls. Németh et al. [15] used an iterative boundary smoothing approach to reduce the set of quench points. Recently, Arcelli et al. [16] suggested to a feature-based approach to locate core, relevant and locally convex CMBs as significant ones in skeletonization. The quench points, i.e., the locations of colliding fire-fronts, have been wellexplored in the context of skeletonization in the form of CMBs. However, the measure of collision impact of meeting fire-fronts at quench points has been surprisingly overlooked in both continuous and digital approaches of skeletonization. In this paper, we formulate a new theoretical framework to characterize the significance of a quench point using the collision impact of fire-fronts and explore its role in filtering noisy quench points. The proposed algorithm is uniformly applicable to both binary and fuzzy objects. It uses local characterization of surface and curve quench points to determine the appropriate support kernels and to compute the average collision impact over the support kernel determining the significance of quench points. The new filtering algorithm has been applied to three-dimensional (3-D) binary and fuzzy objects and its performance under different levels of noise and fuzziness is examined. Also, the performance of this is compared with other DT-based methods of distinguishing between noisy and significant quench points [17-20].
2
Theory and Algorithms
In this section, we define the collision impact and describe the intuitive idea behind its relation with skeletal features of an object in the continuous space. A simple expression of collision impact is presented for digital objects. Finally, a filtering algorithm is described using the measure of collision impact to eliminate noisy quench points while preserving those associated with core shape features of the object. 2.1
Collision Impact and Its Relations with Skeletal Features
Distance transform (DT) [8,9,12] defines the time when a fire-front reaches at a given point during Blum’s grassfire propagation, and a level set of DT gives a snapshot of the entire fire-front at one time instance. Note that the DT function is not differentiable everywhere (e.g., it is not differentiable at ridge points), but it is
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semi-differentiable. Thus, we can compute one-sided directional derivative of DT as follows: ∆· ∆
lim
∆
,
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Fig. 1. Collision-impacts at diffferent skeletal points during grassfire propagation on binary shaapes. (a) The fire-fronts make head-on collision at the point with the maximum collision-impacct of ‘1’. At the point , the fire-fronts collide obliquely generating a weaker collision-impact. (b) The collision impact along a skeletaal branch, originated from a polygonal vertex with a small inteerior angle, e.g., , is higher thaan that along a skeletal branch generated from a vertex with a laarge interior angle, e.g., . (c) The collision-impact along the skeletal branch-segment cconnecting a small protruding struccture to the central skeleton, shown by the dotted line, is low.
where is a direction vector. The uniform speed assumption of Blum’s graassfire propagation leads to th he following equality for the speed function at a poinnt through which a fire-front passes: p max
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1.
The above equality is violaated only at singular or quench points where multiple ffirefronts collide forming skeeletal structures. Although, colliding fire-fronts stopp at quench points, their collisio on strength or impact may vary depending upon the anngle between them. The collision impact of colliding fire-fronts at a point in a binnary object is defined as follows: 1
1
max
,
3
where the function returns the value of if 0 and ‘0’ otherwise. The intuitive idea behind d the formulation of collision impact is explained in Fig. 1 in two-dimension (2-D). Consiider the octagonal shape of Fig. 1 (a) and the head-on coollision of fire-fronts at the poin nt . At the vicinity of , since, there is no point with its DT value greater than that of , the maximum value of is zero. Thus, the collission impact takes the higheest-possible value of ‘1’. Now, let us consider the situatioon at the point where the fire-frronts collide obliquely. Although, the colliding fire-frontss are stopped at , there are increeasing DT values at its vicinity. It can be shown that the m maximum value of is achieved along the direction lying on the tangent sppace of the skeleton at point . Since, along has a finite positive value, the collision impact 1. As shown in Fig. 1 (b), for a polygonal shape, the collission impact along a skeletal brancch originated from a vertex with a small interior angle is large as compared to thee collision impact along a skeletal branch originated from ma
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vertex with a large interiorr angle . Another important observation on collisiionimpact is illustrated in Fig. 1 (c). The collision-impact along the skeletal branch-segm ment on the skeletal branch connecting c a small protrusion to the central skeletal brancch is low. Thus, collision impacct assigns a significance measure to individual skeletall or quench points, and an effecttive algorithms for filtering noisy quench points using coollision impact is presented in Section S 2.3.
Fig. 2. (a,b) Illustration of su urface- (a) and curve-like (b) quench points. (c) Two exam mple support kernels to filter a surfface quench voxel. Voxels colored in green are used for averrage collision impact computation. Four geometrically similar support kernels are constructed ffrom each of these two examples.
2.2
Collision Impact fo or Digital Objects
In this section, we present a generalized formulation quench points that is applicaable to both fuzzy and binary dig gital objects. Here, a 3-D cubic grid, denoted as , whhere is the set of integers, is used as the image space. Each grid element , , is referred d to as a voxel. Traditional definitions of α-adjacent orr αneighborhood [1] between two voxels , , where 6,18,26 , are follow wed in this paper. is useed to denote the set of 26-neighbors of a voxel includding itself while u for the set of all voxels of is used excluding the ccentral voxel . Moreover, thee traditional definitions of α-path, α-connectedness, andd αcomponents [1] are followeed in this paper. A fuzzy digital object , , is a fuzzy set of , where : 0,1 is 0 is its support. Here, 26the membership function and | adjacency is used for objecct voxels in , while 6-adjacency is used for backgrouund voxels, i.e., voxels in . A binary object is defined similarly except tthat the membership function : 0,1 takes the value of ‘1’ for object voxels and ‘0’ for background voxels. A voxel is a CMB in a fuzzy object , if the follow wing [10], inequality holds for every 1 2
|
|.
4
Following the formulation of the collision impact in the continuous space in Eq. 3, the collision impact at any voxel in fuzzy object , denoted by ξ , is defined as follows: ξ
1
max
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|
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The Filtering Algorrithm for Noisy Quench Voxels
The new algorithm for filterring noisy quench points works in the following steps – (1) detect surface- and curve-ty ype quench voxels, (2) determine support kernels for eeach quench voxel depending on o its type, and (3) analyze the collision impact over the support kernels and determ mine the significance of the quench voxel, and (4) rem move quench voxels with their significance measure falling below a predefined threshoold. These steps are described in n the following. Both surface- and curve--quench points may form in 3-D (Fig. 2). A surface queench point is formed when two op pposite fire fronts meet, while a curve quench point is form med when co-planar fire fronts meet m from all directions. In the digital space, a surface-queench voxel is formed by oppositee fire fronts along x-, y- or z-direction and a curve-queench voxel is formed by fire frontts meeting from all eight directions on xy-, yz-, or zx-plan anes. See [13,21] for formal definiitions of surface and curve quench voxels.
Fig. 3. Results of the collision n impact and filtering method on 3-D objects. The top row shhows the original binary objects, wh hile the second and third rows show the initial and filtered queench voxels with color-coded colliision impacts. The fourth and fifth rows present the initial and filtered quench voxels in fuzzy y objects generated by down-sampling.
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Fig. 3. (Continued)
To determine the significcance of a surface-type quench voxel, a support kernel oon a 3 3 digital surface orthog gonal to the surface-normal direction is constructed and the significance is determined as the average collision impact over the support kerrnel pe quench voxel with the horizontal cutting plane, its siggni(Fig. 2 (c)). For a curve-typ ficance is defined as the maaximum collision impact over the support kernels of 3 3 digital surfaces on either siide of the cutting plane. These processes are defined in the following. Let , , be an x-surface quench voxel. To compute the ssupport for , first, a projectiion of three voxels 1, , , , , , , , , 1, , , for some , 1,0,1 , is coom, as follows: puted to generate a 3 3 field f of significance map ,
max ξ
,
,ξ
,
,ξ
,
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The average significancee value over each of eight different suppport nels | 1, , 8 (see Fig. F 2 (c)) is computed. An x-surface-quench voxel is referred to as x-significan nt surface-quench voxel, if any of the average vallues | 1, , 8 is greater than a preset threshold. A voxel is referred to as a ssigxel if it is an x-, y-, or z-significant surface-quench vo xel. nificant surface-quench vox An xy-curve-quench voxell , , is an xy-significant curve-quench vooxel if the largest collision imp pact value in either of the two 3 3 planar cliques , , 1 | , 1,0,1 and , , 1 | , 1,0,1 is greater than a preset threshold. An xy-, yz-, or zx-significant currvequench voxel is referred to as a significant curve-quench voxel. A significant surfaacer to as a significant quench voxel. In this papeer, a or curve-quench voxel is referred constant threshold of 0.5 an nd 0.75 are used for the significance of surface- and currvequench voxels, respectively y.
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Experiments and Results
Results of the filtering algo orithm on a 3-D shape of dinosaur are presented in Figg. 4. Online shapes were construccted at 512×512×512 arrays, which were down-sampled bby a window of 4×4×4 voxels to o generate test phantoms. The binary objects (top row) w were generated by thresholding test t phantoms at 0.5, while the test phantoms were direectly used as fuzzy objects (not sh hown in the figure). The initial set of quench voxels for binnary objects with color-coded co ollision impact values (blue = 0.0, cyan = 0.25, green = 0.5, yellow = 0.75, and red = 1..0) are illustrated on the second row of the figure, while the filtered quench voxels are shown s on the third row. It is observed that that the filterring algorithm has removed visu ually evident noisy quench voxels while preserving the oones capturing the core skeletal shapes of individual objects. Initial quench voxels and the filtered ones for fuzzy objeccts are presented on the last two rows. It is observed thatt the initial sets of quench voxelss for fuzzy objects are larger than that of binary objects. Despite the additional initial quench q voxels for fuzzy objects, the filtering algorithm pproduced satisfactory results. The T visual agreement among the filtered significant queench voxels for binary and fuzzy objects o is highly encouraging that suggests that the algoritthm is robust in the presence of partial p voluming in fuzzy objects.
Fig. 4. An example of the collission impact and the filtering results on a 3-D shape. (a) The origginal binary object; (b) initial quench h voxels with color-coded collision impact; (c,d) quench voxels aafter thresholding at the collision imp mpact using of 0.6 (c) and 0.7 (d); (e) significant quench voxels aafter filtering; (f) final skeleton.
Results of the filtering allgorithm on a 3-D shape of dinosaur are presented in Figg. 3 and its performance is co ompared with simple thresholding on collision imppact. Although, a thresholding on o collision impact partially works in the sense that m most
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Fig. 5. Results of the collisio on impact and the filtering algorithm on dinosaur shape at two down-samplings of 3×3×3 an nd 5×5×5 voxels. (a, c) 2-D slice showing the distributionn of quench voxels along with theirr collision impacts values. (b, d) 3-D results of the filtered queench voxels using the proposed metthod.
Fig. 6. Results of the collisio on impact and filtering algorithm on dinosaur shape under two levels of membership noises, at SNR24 (top row) and SNR6 (bottom row), respectively. L Left om noisy object. Middle column shows the distribution of queench column shows a 2-D slices fro voxels along with their collisio on impact values. Right column displays the result of the filteered quench voxels by the proposed d method in 3-D.
peripheral and noisy quencch voxels are removed and most of the core quench voxxels are preserved, the perform mance is still suboptimal. At the threshold of 0.6, seveeral isolated noisy quench voxeels have survived, while an over-deletion of quench voxxels has occurred at the thresho old of 0.7. On the other hand, the filtering algorithm has removed all visually eviden nt noisy quench voxels while avoiding over-deletion. Finnally, note that the quench vox xels get connected in the final skeleton due to the topoloogy preservation criterion [22,23] during a thinning process. The performance of the filltering algorithm under differeent levels of fuzziness and membership noise are presennted
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in Fig. 5 and Fig. 6. Initial quench voxels on an image slice for two fuzzy objects at 3×3×3 and 5×5×5 down-sampling are presented in Fig. 5a and c. The filtered quench voxels for the two fuzzy objects are shown in Fig. 5b and d. No visually apparent difference in the initial sets of quench voxels is observed. The filtered set of quench voxels at two different levels of fuzziness is visually satisfactory. To study the behavior of the quench voxel generation and the filtering algorithm under membership noise, two fuzzy objects were generated at 3×3×3 down-sampling and then adding white Gaussian noise at signal to noise ratio (SNR) of 24 and 6 (top and bottom rows of Fig. 6, respectively). Visual difference in initial sets of quench voxels are observed at two different levels of noise. Despite the presence of high membership noise, the filtering method successfully eliminated noisy quench voxels while preserving the significant ones. The performance of the algorithm under different levels of boundary noise is presented in Fig. 7. Three images were generated by randomly adding noisy balls of radius one, two, and three voxels. The sets of quench voxels after thresholding at collision impact values of 0.6 and 0.7 are shown on the second and the third rows, respectively. The sets of filtered quench voxels are presented on the last row. Due to boundary noise, several noisy quench voxels survived even after thresholding at a high value of 0.7 for collision impact. In contrast, the filtering algorithm has successfully removed noisy quench voxels, while preserving the core ones. Finally, as observed from Fig. 5 to Fig. 7, the filtered set of quench is visually similar and stable at wide ranges of boundary noise, down-sampling as well as membership noise. It further enforces the validity of the principle of our noisy quench voxel filtering algorithm. The performance of the method at different boundary noise levels is compared with the performance of three existing DT-based methods [17-20] on distinguishing between noisy and significant quench voxels. Gagvani and Silver [17] used the difference of the DT value of a quench voxel from the average DT value of its neighbors as its significance. Siddiqi et al.[20] computed the average outward flux of the DT gradient field as a measure of significance of a quench voxel. Shah [18,19] used the largest angle between incoming fire-fronts as the significance of quench voxels. Results of applications of the three methods are presented in Fig. 8. Gagvani and Silver’s method failed to remove several visually noisy quench voxels while discontinuities on meaningful skeletal segments become apparent. Shah’s method fail to locate quench voxels in the neck, tail, or the legs of the dinosaur. The performance of Siddiqi et al. is more comparable to ours at low noise. However, the performance deteriorated at higher levels of noise where it failed to clean a significant number of noisy voxels.
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Fig. 7. Results of the collision impact and the filtering algorithm on a binary dinosaur shhape under three levels of boundarry noises. The top row shows the 3-D volume of binary objeects. The second and third row show w the quench voxels thresholded at collision impact values off 0.6 and 0.7, respectively. The bo ottom row displays the filtered quench voxels by the propoosed method.
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Fig. 8. Results of applications of the DT-based noisy quench voxel filtering methods by Gagvani and Silver [17] (top row), Siddiqi et al. [20] (middle row), and Shah [18,19] (bottom row) on the binary dinosaur shape under three different levels of boundary noise used in Fig. 7.
4
Conclusions
This paper has presented a new theoretical framework to characterize the significance of a quench point using the collision impact of Blum’s grassfire-fronts. Its role in filtering noisy quench points, prevalent in the skeletonization of real objects, has been explored. Experimental results have demonstrated the effectiveness of method in removing noisy quench voxels while preserving the significant ones capturing core shape features in both binary and fuzzy objects. Initial results has suggested that the method is stable over wide ranges of boundary noise, down-sampling, and membership noise and generates visually satisfactory results despite significant image artifacts. Our method offers a unified solution for both binary and fuzzy objects, while existing methods are applicable to binary objects, only. Also, the initial results suggests that the current method may perform better as compared to existing ones, especially, at higher levels of noise. Currently, we are conducting quantitative analysis of its performance and exploring its role in improving the results of skeletonization. Acknowledgements. This work was supported by the NIH grant R01-AR054439.
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